To determine the area of sector [tex]\(AOB\)[/tex] given a circle with points [tex]\(A\)[/tex] and [tex]\(B\)[/tex] lying on it, centered at point [tex]\(O\)[/tex] with radius [tex]\(OA = 5\)[/tex], and the arc length [tex]\(\hat{AB}\)[/tex] being [tex]\(\frac{1}{4}\)[/tex] of the circumference of the circle, we can follow these steps:
1. Calculate the circumference of the circle:
The formula for the circumference [tex]\(C\)[/tex] of a circle is given by:
[tex]\[
C = 2 \pi r
\][/tex]
Given [tex]\(\pi = 3.14\)[/tex] and [tex]\(r = 5\)[/tex],
[tex]\[
C = 2 \times 3.14 \times 5 = 31.4
\][/tex]
2. Determine the arc length [tex]\(\hat{AB}\)[/tex]:
The arc length is specified as [tex]\(\frac{1}{4}\)[/tex] of the total circumference. Therefore,
[tex]\[
\text{Arc length} = \frac{1}{4} \times 31.4 = 7.85
\][/tex]
3. Calculate the angle [tex]\(\theta\)[/tex] subtended by the arc at the center (in radians):
The angle in radians for the arc can be found using the proportion of the arc length to the total circumference:
[tex]\[
\theta = \frac{\text{Arc length}}{\text{Circumference}} \times 2\pi
\][/tex]
Substituting the values,
[tex]\[
\theta = \frac{7.85}{31.4} \times 2 \times 3.14 = \frac{1}{4} \times 2 \times 3.14 = 1.57
\][/tex]
4. Calculate the area of the sector [tex]\(AOB\)[/tex]:
The formula for the area of a sector is given by:
[tex]\[
\text{Area of sector} = \frac{1}{2} r^2 \theta
\][/tex]
Given [tex]\(r = 5\)[/tex] and [tex]\(\theta = 1.57\)[/tex],
[tex]\[
\text{Area of sector} = \frac{1}{2} \times 5^2 \times 1.57 = \frac{1}{2} \times 25 \times 1.57 = 19.625
\][/tex]
Thus, the area of sector [tex]\(AOB\)[/tex] is [tex]\(19.625\)[/tex] square units, which is closest to option A when rounded to one decimal place.
The correct answer is:
A. 19.6 square units
